How to show 2-tori is diffeomorphic to S^3

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Discussion Overview

The discussion centers on the relationship between 2-tori and S^3, specifically exploring how 2-tori can be shown to be diffeomorphic to S^3 or embedded within it. The scope includes theoretical considerations and mathematical reasoning related to topology and differential geometry.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant defines 2-tori as the set {(z1,z2)| |z1|=c1,|z2|=c2} and questions how to show it is diffeomorphic to S^3.
  • Another participant asserts that a torus is not diffeomorphic to S^3, introducing a counterpoint to the initial claim.
  • A subsequent post corrects the initial question, suggesting the focus should be on how 2-tori are embedded into S^3.
  • A participant proposes a mapping from R^2 into R^4, specifically (x,y) -> (1/2^.5)(cos x, sin x, cos y, sin y), claiming that the image represents a torus in S^3.
  • Another participant introduces a smooth function F:C^2\{0} to C, defined as F(z1,z2)=z1^p+z2^q with p and q being relatively prime integers greater than or equal to 2, and questions how to show that the intersection of S^3 and F^(-1)(0) is diffeomorphic to 2-tori.

Areas of Agreement / Disagreement

Participants express disagreement regarding the diffeomorphism between 2-tori and S^3, with some proposing embedding instead. The discussion remains unresolved with multiple competing views on the nature of the relationship between these mathematical structures.

Contextual Notes

There are limitations regarding the assumptions made about the definitions of toroidal structures and the conditions under which diffeomorphism or embedding is considered. The mathematical steps and implications of the proposed mappings and functions are not fully resolved.

robforsub
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Define 2-tori as {(z1,z2)| |z1|=c1,|z2|=c2} for c1 and c2 are constants, how to show that it is diffeomorphic to S^3
 
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robforsub said:
Define 2-tori as {(z1,z2)| |z1|=c1,|z2|=c2} for c1 and c2 are constants, how to show that it is diffeomorphic to S^3

A torus is not diffeomorphic to S^3.
 
My mistake, it should be how 2-tori is embedded into S^3
 
robforsub said:
My mistake, it should be how 2-tori is embedded into S^3

map R^2 into R^4 by (x,y) -> (1/2^.5)(cos x, sin x, cos y, sin y)

The image is a torus in S^3
 
What if there is a smooth function F:C^2\{0} to C, defined as F(z1,z2)=z1^p+z2^q with p and q>=2 and they are relatively prime, then how to show that S^3 intersect F^(-1)(0) is diffeomorphic to 2-tori?
 

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